Berry-Esséen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion

2012/03/13 by Soufiane Aazizi, Aazizi, Soufiane, Khalifa Es-Sebaiy +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1203.2786

arxiv created 2012/03/27 · arxiv updated 2012/03/28

Abstract

Let B be a bifractional Brownian motion with parameters H∈ (0, 1) and K∈(0,1]. For any n≥1, set Z_n =∑_i=0^n-1[n^2HK(B_(i+1)/n-B_i/n)²-\E((B_i+1-B_i)²)]. We use the Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati \citeNP09 to derive, in the case when 0<HK≤3/4, Berry-Esséen-type bounds for the Kolmogorov distance between the law of the correct renormalization V_n of Z_n and the standard normal law. Finally, we study almost sure central limit theorems for the sequence V_n.

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